Properties

Label 6004.183
Modulus $6004$
Conductor $6004$
Order $78$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6004, base_ring=CyclotomicField(78))
 
M = H._module
 
chi = DirichletCharacter(H, M([39,65,46]))
 
pari: [g,chi] = znchar(Mod(183,6004))
 

Basic properties

Modulus: \(6004\)
Conductor: \(6004\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(78\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 6004.cy

\(\chi_{6004}(183,\cdot)\) \(\chi_{6004}(411,\cdot)\) \(\chi_{6004}(483,\cdot)\) \(\chi_{6004}(487,\cdot)\) \(\chi_{6004}(863,\cdot)\) \(\chi_{6004}(1699,\cdot)\) \(\chi_{6004}(1703,\cdot)\) \(\chi_{6004}(1927,\cdot)\) \(\chi_{6004}(2231,\cdot)\) \(\chi_{6004}(2311,\cdot)\) \(\chi_{6004}(2611,\cdot)\) \(\chi_{6004}(2767,\cdot)\) \(\chi_{6004}(3447,\cdot)\) \(\chi_{6004}(3527,\cdot)\) \(\chi_{6004}(3679,\cdot)\) \(\chi_{6004}(3755,\cdot)\) \(\chi_{6004}(3903,\cdot)\) \(\chi_{6004}(3907,\cdot)\) \(\chi_{6004}(4055,\cdot)\) \(\chi_{6004}(4435,\cdot)\) \(\chi_{6004}(4587,\cdot)\) \(\chi_{6004}(4891,\cdot)\) \(\chi_{6004}(4895,\cdot)\) \(\chi_{6004}(5579,\cdot)\)

sage: chi.galois_orbit()
 
order = charorder(g,chi)
 
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Related number fields

Field of values: $\Q(\zeta_{39})$
Fixed field: Number field defined by a degree 78 polynomial

Values on generators

\((3003,2529,3953)\) → \((-1,e\left(\frac{5}{6}\right),e\left(\frac{23}{39}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(17\)\(21\)\(23\)
\( \chi_{ 6004 }(183, a) \) \(1\)\(1\)\(e\left(\frac{12}{13}\right)\)\(e\left(\frac{35}{39}\right)\)\(e\left(\frac{59}{78}\right)\)\(e\left(\frac{11}{13}\right)\)\(e\left(\frac{47}{78}\right)\)\(e\left(\frac{17}{78}\right)\)\(e\left(\frac{32}{39}\right)\)\(e\left(\frac{28}{39}\right)\)\(e\left(\frac{53}{78}\right)\)\(-1\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 6004 }(183,a) \;\) at \(\;a = \) e.g. 2